Finite-Difference Jacobians and Hessians
Contents
Finite-Difference Jacobians and Hessians¶
Randall Romero Aguilar, PhD
This demo is based on the original Matlab demo accompanying the Computational Economics and Finance 2001 textbook by Mario Miranda and Paul Fackler.
Original (Matlab) CompEcon file: demdif05.m
Running this file requires the Python version of CompEcon. This can be installed with pip by running
!pip install compecon --upgrade
Last updated: 2021-Oct-01
Initial tasks¶
from compecon import jacobian, hessian
import numpy as np
np.set_printoptions(precision=15)
Example 1¶
The exact Jacobian of
\[\begin{equation*}
f(x_1,x_2) = \begin{bmatrix}\exp(x_1)-x_2 \\ x_1+x_2^2 \\ (1-x_1)\log(x_2)\end{bmatrix}
\end{equation*}\]
at \((0,1)\) is
\[\begin{equation*}
f'(x_1,x_2) = \begin{bmatrix}1 & -1 \\ 1 & 2 \\ 0 & 1\end{bmatrix}
\end{equation*}\]
def f(x):
x1, x2 = x
y = [np.exp(x1)-x2,
x1 + x2**2,
(1-x1)*np.log(x2)]
return np.array(y)
jacobian(f, [0, 1])
array([[ 1.000000000014386, -1. ],
[ 0.999999999996052, 1.999999999990833],
[ 0. , 1.000000000012223]])
Ejemple 2¶
The exact Hessian of
\[\begin{equation*}
f(x_1,x_2) = x_1^2 \exp(-x_2)
\end{equation*}\]
at \((1,0)\) is
\[\begin{equation*}
f''(x_1,x_2) = \begin{bmatrix}2 & -2 \\ -2 & 1\end{bmatrix}.
\end{equation*}\]
def f(x):
x1, x2 = x
return x1**2 * np.exp(-x2)
hessian(f,[1, 0])
array([[ 2. , -2.000000006519258],
[-2.000000006519258, 0.999999985098839]])